cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

A026642 a(n) = A026637(2*n-1, n-2).

Original entry on oeis.org

1, 7, 28, 112, 439, 1711, 6652, 25846, 100450, 390670, 1520764, 5925718, 23112931, 90239407, 352654084, 1379410438, 5400188206, 21157958962, 82959736504, 325514137048, 1278093308806, 5021436970822, 19740128055928
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026637, A026638, A026639, A026640, A026641, A026643, A026644.
Cf. A026645, A026646, A026647, A026966, A026967, A026968, A026969.
Cf. A026970.

Programs

  • Magma
    [n le 2 select 7^(n-1) else ((7*n^2+3*n+2)*Self(n-1) + 2*n*(2*n+1)*Self(n-2))/(2*(n-1)*(n+2)): n in [1..40]]; // G. C. Greubel, Jul 01 2024
  • Mathematica
    a[n_]:= a[n]= If[n<4, 7^(n-2), ((7*n^2-11*n+6)*a[n-1] + 2*(n-1)*(2*n- 1)*a[n-2])/(2*(n-2)*(n+1))];
Table[a[n], {n,2,40}] (* G. C. Greubel, Jul 01 2024 *)
  • SageMath
    @CachedFunction
def a(n): # a = A026642
    if n<4: return 7^(n-2)
    else: return ((7*n^2-11*n+6)*a(n-1) + 2*(n-1)*(2*n-1)*a(n-2))/(2*(n-2)*(n+1))
[a(n) for n in range(2,41)] # G. C. Greubel, Jul 01 2024

Formula

a(n) = ( (7*n^2 - 11*n + 6)*a(n-1) + 2*(n-1)*(2*n-1)*a(n-2) )/(2*(n-2)*(n+1)), n >= 4. - G. C. Greubel, Jul 01 2024

A026638 a(n) = A026637(2*n, n).

Original entry on oeis.org

1, 2, 8, 26, 92, 332, 1220, 4538, 17036, 64412, 244928, 935684, 3588392, 13806704, 53271548, 206040506, 798600332, 3101109164, 12062148368, 46986821516, 183276382472, 715748620424, 2798274135368, 10951009023716, 42895901012792, 168167959150232, 659793819847040
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026637, A026639, A026640, A026642, A026643, A026644, A026645.
Cf. A026646, A026647, A026966, A026967, A026968, A026969, A026970.

Programs

  • Magma
    [1] cat [n le 2 select 2^(2*n-1) else ((7*n-4)*Self(n-1) + 2*(2*n-1)*Self(n-2))/(2*n): n in [1..40]]; // G. C. Greubel, Jul 01 2024
  • Mathematica
    CoefficientList[Series[1/(2+x)+3/((2+x)*Sqrt[1-4*x])-1,{x,0,20}],x] (* Vaclav Kotesovec, Oct 21 2012 *)
  • PARI
    my(x='x+O('x^66)); Vec( 1/(2+x)+3/((2+x)*sqrt(1-4*x))-1 ) \\ Joerg Arndt, May 04 2013
  • SageMath
    @CachedFunction
def a(n): # a = A026638
    if n<3: return 2^(n*(n+1)/2)
    else: return ((7*n-4)*a(n-1) + 2*(2*n-1)*a(n-2))/(2*n)
[a(n) for n in range(41)] # G. C. Greubel, Jul 01 2024

Formula

From Vaclav Kotesovec, Oct 21 2012: (Start)
G.f.: (3 - (x+1)*sqrt(1-4*x))/((x+2)*sqrt(1-4*x)).
Recurrence: 2*n*a(n) = (7*n-4)*a(n-1) + 2*(2*n-1)*a(n-2).
a(n) ~ 2^(2*n+2)/(3*sqrt(Pi*n)) (End)

A026639 a(n) = A026637(2*n, n-1).

Original entry on oeis.org

1, 5, 20, 74, 278, 1049, 3980, 15170, 58052, 222914, 858512, 3314960, 12829070, 49748705, 193259660, 751954250, 2929965020, 11431262390, 44651369720, 174597927740, 683388447260, 2677230376490, 10496941482680, 41188078562324
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026637, A026638, A026640, A026641, A026642, A026643, A026644.
Cf. A026645, A026646, A026647, A026966, A026967, A026968, A026969.
Cf. A026970.

Programs

  • Magma
    [1] cat [n le 2 select 5*(3*n-2) else ((7*n^2+10*n+4)*Self(n-1) + 2*(2*n+1)*(n+1)*Self(n-2))/(2*n*(n+2)): n in [1..40]]; // G. C. Greubel, Jul 01 2024
  • Mathematica
    a[n_]:= a[n]= If[n<4, (5*4^(n-1) -Boole[n==1])/4, ((7*n^2-4*n+1)*a[n- 1] +2*n*(2*n-1)*a[n-2])/(2*(n^2-1))];
Table[a[n], {n,40}] (* G. C. Greubel, Jul 01 2024 *)
  • SageMath
    @CachedFunction
def a(n): # a = A026639
    if n<4: return (5*4^(n-1) - 0^(n-1))/4
    else: return ((7*n^2 - 4*n + 1)*a(n-1) + 2*n*(2*n-1)*a(n-2))/(2*(n^2-1))
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 01 2024

Formula

a(n) = ((7*n^2 - 4*n + 1)*a(n-1) + 2*n*(2*n-1)*a(n-2))/(2*(n^2-1)), with a(0) = 1, a(1) = 5, a(2) = 20. - G. C. Greubel, Jul 01 2024

A026640 a(n) = A026637(2*n, n-2).

Original entry on oeis.org

1, 8, 38, 161, 662, 2672, 10676, 42398, 167756, 662252, 2610758, 10283861, 40490702, 159394424, 627456188, 2470223186, 9726696572, 38308366784, 150916209308, 594704861546, 2344206594332, 9243186573248, 36456892635848
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026637, A026638, A026639, A026641, A026642, A026643, A026644.
Cf. A026645, A026646, A026647, A026966, A026967, A026968, A026969.
Cf. A026970.

Programs

  • Magma
    [1] cat [n le 2 select 4^(n+1) -3^(n+1) +1 else ((7*n^2+24*n+24 )*Self(n-1) + 2*(2*n+3)*(n+2)*Self(n-2))/(2*n*(n+4)): n in [1..40]]; // G. C. Greubel, Jul 01 2024
  • Mathematica
    a[n_]:= a[n]= If[n<5, 4^(n-1) -3^(n-1) +1 -Boole[n==2], ((7*n^2 -4*n + 4)*a[n-1] +2*n*(2*n-1)*a[n-2])/(2*(n-2)*(n+2))];
Table[a[n], {n,2,40}] (* G. C. Greubel, Jul 01 2024 *)
  • SageMath
    @CachedFunction
def a(n): # a = A026640
    if n<5: return 4^(n-1) -3^(n-1) +1 -int(n==2)
    else: return ((7*n^2-4*n+4)*a(n-1) + 2*n*(2*n-1)*a(n-2))/(2*(n-2)*(n+2))
[a(n) for n in range(2,41)] # G. C. Greubel, Jul 01 2024

Formula

a(n) = ((7*n^2 - 4*n + 4)*a(n-1) + 2*n*(2*n-1)*a(n-2))/(2*(n-2)*(n+2)), n >= 5. - G. C. Greubel, Jul 01 2024

A026643 a(n) = A026637(n, floor(n/2)).

Original entry on oeis.org

1, 1, 2, 4, 8, 13, 26, 46, 92, 166, 332, 610, 1220, 2269, 4538, 8518, 17036, 32206, 64412, 122464, 244928, 467842, 935684, 1794196, 3588392, 6903352, 13806704, 26635774, 53271548, 103020253, 206040506, 399300166, 798600332
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026637, A026638, A026639, A026640, A026641, A026642, A026644.
Cf. A026966, A026967, A026968, A026969, A026970.

Programs

  • Magma
    [1] cat [n le 4 select 2^(n-1) else (4*Self(n-1) +(7*n-9)*Self(n-2) +2*Self(n-3) +4*(n-1)*Self(n-4))/(2*(n+1)): n in [1..40]]; // G. C. Greubel, Jul 01 2024
  • Mathematica
    T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[k==1 || k==n-1, Floor[(3*n -1)/2], T[n-1,k-1] + T[n-1,k] ]]; (* A026637 *)
A026643[n_]:= T[n, Floor[n/2]];
Table[A026643[n], {n,0,40}] (* G. C. Greubel, Jul 01 2024 *)
  • SageMath
    @CachedFunction
def a(n): # a = A026643
    if n<5: return (1,1,2,4,8)[n]
    else: return (4*a(n-1) +(7*n-9)*a(n-2) +2*a(n-3) +4*(n-1)*a(n-4))/(2*(n+1))
[a(n) for n in range(41)] # G. C. Greubel, Jul 01 2024

Formula

a(n) = (4*a(n-1) + (7*n-9)*a(n-2) + 2*a(n-3) + 4*(n-1)*a(n-4))/(2*(n+1)) with a(0) = a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 8. - G. C. Greubel, Jul 01 2024

A026646 a(n) = Sum_{i=0..n} Sum_{j=0..n} A026637(i,j).

Original entry on oeis.org

1, 3, 7, 17, 37, 79, 163, 333, 673, 1355, 2719, 5449, 10909, 21831, 43675, 87365, 174745, 349507, 699031, 1398081, 2796181, 5592383, 11184787, 22369597, 44739217, 89478459, 178956943, 357913913, 715827853, 1431655735
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

a(n) indexes the corner blocks on the Jacobsthal spiral built from blocks of unit area (using J(1) and J(2) as the sides of the first block). - Paul Barry, Mar 06 2008
Partial sums of A026644, which are the row sums of A026637. - Paul Barry, Mar 06 2008

Links

Crossrefs

Cf. A000126, A001045, A026637, A026644, A084639, A109613.
Cf. A026637, A026638, A026639, A026640, A026641, A026642, A026643.
Cf. A026644, A026966, A026967, A026968, A026969, A026970.

Programs

  • Magma
    [(2^(n+4) -(6*n+9+(-1)^n))/6: n in [0..40]]; // G. C. Greubel, Jul 01 2024
  • Mathematica
    CoefficientList[Series[(1-x^2+2x^3)/((1-x)(1-2x)(1-x^2)), {x, 0, 29}], x] (* Metin Sariyar, Sep 22 2019 *)
LinearRecurrence[{3,-1,-3,2}, {1,3,7,17}, 41] (* G. C. Greubel, Jul 01 2024 *)
  • SageMath
    [(2^(n+4) - (-1)^n -9 - 6*n)/6 for n in range(41)] # G. C. Greubel, Jul 01 2024

Formula

G.f.: (1 -x^2 +2*x^3)/((1-x)*(1-2*x)*(1-x^2)). - Ralf Stephan, Apr 30 2004
From Paul Barry, Mar 06 2008: (Start)
a(n) = A001045(n+3) - 2*floor((n+2)/2).
a(n) = -n + Sum_{k=0..n} A001045(k+2) = A084639(n+1) - n. (End)
a(n+1) = 2*a(n) + A109613(n), a(0)=1. - Paul Curtz, Sep 22 2019

A026966 Self-convolution of array T given by A026637.

Original entry on oeis.org

1, 2, 6, 34, 116, 438, 1606, 6002, 22548, 85342, 324716, 1241166, 4762000, 18329054, 70742790, 273689362, 1061055812, 4121135838, 16032654628, 62463904958, 243682621224, 951781620022, 3721520393036, 14565753339054, 57060974849656, 223721451300518, 877830004120296
Offset: 0

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026967 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026637.

Original entry on oeis.org

1, 4, 24, 90, 365, 1376, 5272, 20098, 77016, 295836, 1139834, 4402490, 17042341, 66101848, 256838544, 999512058, 3895191476, 15199297796, 59377405654, 232208188726, 908978576496, 3561356276424, 13964734085774, 54799750800530, 215193550759630, 845596666261036
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026968 a(n) = Sum_{k=0..n-2} T(n,k) * T(n,k+2), with T given by A026637.

Original entry on oeis.org

1, 8, 41, 208, 856, 3552, 14172, 56462, 223292, 881656, 3475141, 13688112, 53891760, 212145608, 835111160, 3287779006, 12946154944, 50989472460, 200879577216, 791617503410, 3120501139160, 12304529289928, 48533074308046, 191488113441258, 755744827552836, 2983563960355832
Offset: 2

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026969 a(n) = Sum_{k=0..n-3} T(n,k) * T(n,k+3), with T given by A026637.

Original entry on oeis.org

1, 10, 75, 372, 1796, 7796, 33352, 138790, 571995, 2334558, 9473195, 38258984, 153981400, 618042736, 2475405456, 9897862062, 39523226642, 157650156506, 628287555844, 2502170730172, 9959281642416, 39622214018100, 157575049910690, 626477638980782, 2490112612092175
Offset: 3

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 20 2019
Showing 1-10 of 16 results. Next

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