A074377 Generalized 10-gonal numbers: m*(4*m - 3) for m = 0, +- 1, +- 2, +- 3, ...

0, 1, 7, 10, 22, 27, 45, 52, 76, 85, 115, 126, 162, 175, 217, 232, 280, 297, 351, 370, 430, 451, 517, 540, 612, 637, 715, 742, 826, 855, 945, 976, 1072, 1105, 1207, 1242, 1350, 1387, 1501, 1540, 1660, 1701, 1827, 1870, 2002, 2047, 2185, 2232, 2376, 2425

Offset: 0

Author

W. Neville Holmes, Sep 04 2002

Keywords

• nonn
• easy

Comments

Also called generalized decagonal numbers.

Odd triangular numbers decremented and halved.

It appears that this is zero together with the partial sums of A165998. - Omar E. Pol, Sep 10 2011 [this is correct, see the g.f., Joerg Arndt, Sep 29 2013]

Also, A033954 and positive members of A001107 interleaved. - Omar E. Pol, Aug 04 2012

Also, numbers m such that 16*m+9 is a square. After 1, therefore, there are no squares in this sequence. - Bruno Berselli, Jan 07 2016

Convolution of the sequences A047522 and A059841. - Ilya Gutkovskiy, Mar 16 2017

Numbers k such that the concatenation k5625 is a square. - Bruno Berselli, Nov 07 2018

Exponents in expansion of Product_{n >= 1} (1 + x^(8*n-7))*(1 + x^(8*n-1))*(1 - x^(8*n)) = 1 + x + x^7 + x^10 + x^22 + .... - Peter Bala, Dec 10 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Neville Holmes, More Gemometric Integer Sequences. [Wayback Machine copy]
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A011848, A014493, A074378, A118277, A165998.

Cf. A001107 (10-gonal numbers).

Column 6 of A195152.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), this sequence (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

Programs

Magma

[n^2+n-1/4+(-1)^n/4+n*(-1)^n/2: n in [0..50]]; // Vincenzo Librandi, Sep 29 2013

Mathematica

CoefficientList[Series[x(1 +6x +x^2)/((1-x)(1-x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 29 2013 *)
LinearRecurrence[{1,2,-2,-1,1}, {0,1,7,10,22}, 50] (* G. C. Greubel, Nov 07 2018 *)

PARI

a(n)=(2*n+3-4*(n%2))*(n-n\2)

PARI

concat([0],Vec(x*(1 + 6*x + x^2)/((1 - x)*(1 - x^2)^2) +O(x^50))) \\ Indranil Ghosh, Mar 16 2017

Python

def A074377(n): return (n+1>>1)*((n<<1)+(-1 if n&1 else 3)) # Chai Wah Wu, Mar 11 2025

Formula

(n(n+1)-2)/4 where n(n+1)/2 is odd.

G.f.: x*(1+6*x+x^2)/((1-x)*(1-x^2)^2). - Michael Somos, Mar 04 2003

a(2*k) = k*(4*k+3); a(2*k+1) = (2*k+1)^2+k. - Benoit Jubin, Feb 05 2009

a(n) = n^2+n-1/4+(-1)^n/4+n*(-1)^n/2. - R. J. Mathar, Oct 08 2011

Sum_{n>=1} 1/a(n) = (4 + 3*Pi)/9. - Vaclav Kotesovec, Oct 05 2016

E.g.f.: exp(x)*x^2 + (2*exp(x) - exp(-x)/2)*x - sinh(x)/2. - Ilya Gutkovskiy, Mar 16 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 4/9. - Amiram Eldar, Feb 28 2022

a(n) = (n+1)*(2*n-1)/2 if n is odd and a(n) = n*(2*n+3)/2 if n is even. - Chai Wah Wu, Mar 11 2025

Extensions

New name from T. D. Noe, Apr 21 2006

Formula in sequence name from Omar E. Pol, May 28 2012