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 1-4 of 4 results.

A129359 Numbers k such that A129357(8*k) == 2 (mod 4).

Original entry on oeis.org

1, 3, 5, 6, 8, 9, 13, 15, 16, 19, 20, 23, 26, 27, 28, 29, 31, 33, 34, 35, 36, 38, 45, 48, 50, 51, 53, 55, 56, 59, 61, 63, 64, 69, 71, 73, 77, 78, 83, 84, 85, 86, 89, 91, 93, 94, 96, 100, 101, 103, 104, 108, 110, 115, 119, 121, 124, 127, 129, 131, 133, 134, 135
Offset: 1

Views

Author

Paul D. Hanna, Apr 11 2007

Keywords

Crossrefs

Cf. A129357.

Programs

  • PARI
    lista(nn) = my(y=prod(k=1, 8*nn, (1-x)^4*sum(j=1, k, binomial(j+2, 3)*x^(j-1)) + x*O(x^(8*nn)))); for(k=1, nn, if(polcoeff(y, 8*k)%4 == 2, print1(k, ", "))); \\ Jinyuan Wang, Jan 01 2021

Extensions

More terms from, name edited and offset changed to 1 by Jinyuan Wang, Jan 01 2021

A129355 G.f.: A(x) = Product_{n>=1} [ (1-x)^2*(1 + 2x + 3x^2 +...+ n*x^(n-1)) ].

Original entry on oeis.org

1, -2, -2, 4, -1, 12, -26, 38, -51, 6, 98, -190, 138, 60, 132, -1296, 2990, -3738, 3350, -3752, 4077, 1194, -12272, 18528, -14848, 9018, -2002, 5644, -86729, 290596, -514158, 611070, -603150, 657792, -952808, 1406568, -1208636, -635286, 3507362, -5062866, 3791614
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2007

Keywords

Comments

a(k) == 1 (mod 2) at k = 4*A001318(n) for n>=0, where A001318 are the generalized pentagonal numbers: m(3m-1)/2, m=0,+-1,+-2,....

Examples

			A(x) = (1 - 2x + x^2)(1 - 3x^2 + 2x^3)(1 - 4x^3 + 3x^4)(1 - 5x^4 + 4x^5)*...
Terms are even except at positions given by:
a(n) == 1 (mod 2) at n = [0, 4, 8, 20, 28, 48, 60, 88,...,4*A001318(n),...].

Crossrefs

Cf. A129356, A129357, A129358.

Programs

  • PARI
    a(n)=if(n==0,1,polcoeff(prod(k=1,n,1-(k+1)*x^k+k*x^(k+1)+x*O(x^n)),n))

Formula

G.f.: A(x) = Product_{n>=1} ( 1 - (n+1)*x^n + n*x^(n+1) ) . G.f.: A(x) = Product_{n>=1} [ (1-x)*(1 + x + x^2 +...+ x^(n-1) - n*x^n) ] .

A129356 G.f.: A(x) = Product_{n>=1} [ (1-x)^3*(1 + 3x + 6x^2 +...+ n(n+1)/2*x^(n-1)) ].

Original entry on oeis.org

1, -3, -3, 15, -15, 66, -261, 618, -1155, 1040, 2361, -11616, 23733, -27027, 29394, -132318, 545790, -1383459, 2418896, -3383679, 4278462, -3127320, -8332866, 42021990, -99069516, 160683318, -200247795, 214883010, -345461022, 1184850729, -3966311448, 9899287254, -18787986009
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2007

Keywords

Comments

a(k) != 0 (mod 3) at k = 9*A001318(n) for n>=0, where A001318 are the generalized pentagonal numbers: m(3m-1)/2, m=0,+-1,+-2,...; a(k) == 1 (mod 3) at k = 9*A036498(n) (n>=0); a(k) == -1 (mod 3) at k = 9*A036499(n) (n>=0).

Examples

			A(x) = (1-3x+3x^2-x^3)(1-6x^2+8x^3-3x^4)(1-10x^3+15x^4-6x^5)*...
*( 1 - (n+1)(n+2)/2*x^n + n(n+2)*x^(n+1) - n(n+1)/2*x^(n+2) )*...
Terms are divisible by 3 except at positions given by:
a(n) == 1 (mod 3) at n = [0, 45, 63, 198, 234, 459,...,9*A036498(k),..];
a(n) == -1 (mod 3) at n = [9, 18, 108, 135, 315, 360,..,9*A036499(k),..].

Crossrefs

Cf. A129355, A129357, A129358; A001318, A036498, A036499.

Programs

  • PARI
    {a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-x)^3*sum(j=1,k,j*(j+1)/2*x^(j-1)) +x*O(x^n)),n))}

Formula

G.f.: A(x) = Product_{n>=1} [ 1 - (n+1)(n+2)/2*x^n + n(n+2)*x^(n+1) - n(n+1)/2*x^(n+2) ].

A129358 G.f.: A(x) = Product_{n>=1} [ (1-x)^5*(1 + 5x + 15x^2 +...+ n(n+1)(n+2)(n+3)/4!*x^(n-1)) ].

Original entry on oeis.org

1, -5, -5, 70, -180, 770, -4760, 20840, -68085, 147890, -795, -1679855, 8378195, -25065005, 56439545, -145200415, 612604910, -2764023765, 10020060660, -28723695265, 67618167310, -128945409045, 137921330680, 375948665405, -3167538981120, 12823443150644, -38103903888575
Offset: 0

Views

Author

Paul D. Hanna, Apr 11 2007

Keywords

Comments

a(k) != 0 (mod 5) at k = 25*A001318(n) for n>=0, where A001318 are the generalized pentagonal numbers: m(3m-1)/2, m=0,+-1,+-2,...; a(k) == 1 (mod 5) at k = 25*A036498(n) (n>=0); a(k) == -1 (mod 5) at k = 25*A036499(n) (n>=0).

Examples

			A(x) = (1-5x+10x^2-10x^3+5x^4-x^5)*(1-15x^2+40x^3-45x^4+24x^5-5x^6)*(1-35x^3+105x^4-126x^5+70x^6-15x^7)*(1-70x^4+224x^5-280x^6+160x^7-35x^8)*...
Terms are divisible by 5 except at positions given by 25*A001318(n):
a(n) == 1 (mod 5) at n = [0, 125, 175, 550, 650,...,25*A036498(k),...];
a(n) == -1 (mod 5) at n = [25, 50, 300, 375, 875,...,25*A036499(k),...].

Crossrefs

Cf. A129355, A129356, A129357; A001318, A036498, A036499.

Programs

  • PARI
    {a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-x)^5*sum(j=1,k,binomial(j+3,4)*x^(j-1)) +x*O(x^n)),n))}

Formula

G.f.: A(x) = Product_{n>=1} [ 1 - (n+1)(n+2)(n+3)(n+4)/4!*x^n + 4n(n+2)(n+3)(n+4)/4!*x^(n+1) - 6n(n+1)(n+3)(n+4)/4!*x^(n+2) + 4n(n+1)(n+2)(n+4)/4!*x^(n+3) - n(n+1)(n+2)(n+3)/4!*x^(n+4) ].
Showing 1-4 of 4 results.

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